Combined events - Teacher Slides.pptx

EDEXCEL GCSE MATHS · FOUNDATION & HIGHER

Combined events

Probability

Lesson 1 of 6

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EDEXCEL GCSE MATHS · FOUNDATION & HIGHER

Warm-up

Probability

Lesson 1 of 6

Answer each one, then check.

1

What is the probability of rolling a 3 on a fair dice?

2

What is the probability of getting tails on a fair coin?

3

Simplify 6/36.

4

What do all the probabilities of an event add up to?

5

If P(rain) = 0.3, what is P(not rain)?

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EDEXCEL GCSE MATHS · FOUNDATION & HIGHER

Warm-up - Answers

Probability

Lesson 1 of 6

1

What is the probability of rolling a 3 on a fair dice?

1/6

2

What is the probability of getting tails on a fair coin?

½

3

Simplify 6/36.

1/6

4

What do all the probabilities of an event add up to?

1

5

If P(rain) = 0.3, what is P(not rain)?

0.7

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EDEXCEL GCSE MATHS · FOUNDATION & HIGHER

Learning Objectives

Probability

Lesson 1 of 6

1

Use the probability scale and the formula for equally likely outcomes.

2

List the outcomes of two events systematically.

3

Draw and use a sample space diagram.

4

Work out probabilities of combined events.

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EDEXCEL GCSE MATHS · FOUNDATION & HIGHER

Probability

Lesson 1 of 6

THE FORMULA

For equally likely outcomes, probability is the number of successful outcomes divided by the total number of possible outcomes.

P(event) = (number of ways the event can happen)/(total number of equally likely outcomes)

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Listing Outcomes

Probability

Lesson 1 of 6

A fair coin is flipped and a fair dice is rolled. List all the possible outcomes, and find the probability of a head and a 6.

1

Write each coin outcome with each dice outcome

H1, H2, H3, H4, H5, H6, T1, T2, T3, T4, T5, T6

2

Count the outcomes

12

3

Only one of these is a head with a 6

H6

ANSWER

There are 12 outcomes, so P(head and 6) = 1/12.

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EDEXCEL GCSE MATHS · FOUNDATION & HIGHER

A Sample Space Diagram

Probability

Lesson 1 of 6

A sample space diagram shows every outcome, so nothing is missed.

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EDEXCEL GCSE MATHS · FOUNDATION & HIGHER

Totals from Two Dice

Probability

Lesson 1 of 6

Each cell is the sum of the two dice.

First dice

1

2

3

4

1

2

3

4

5 | 6 | 7

2

3

4

5

6 | 7 | 8

3

4

5

6

7 | 8 | 9

4

5

6

7

8 | 9 | 10

5

6

7

8

9 | 10 | 11

6

7

8

9

10 | 11 | 12

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EDEXCEL GCSE MATHS · FOUNDATION & HIGHER

Using the Table

Probability

Lesson 1 of 6

Two fair dice are rolled and the scores are added. Find (a) P(total is 7) and (b) P(total is at least 10).

1

(a) Count the 7s in the table

6 out of 36

2

(a) Simplify

6/36 = 1/6

3

(b) Count the totals 10, 11 or 12

3 + 2 + 1 = 6

4

(b) Probability

6/36 = 1/6

ANSWER

(a) 1/6 (b) 1/6

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Two Spinners

Probability

Lesson 1 of 6

Spinner A has three equal sections numbered 1, 2, 3. Spinner B has four equal sections numbered 1, 2, 3, 4. Both are spun and the scores added. Find the probability that the total is 5, and that it is even.

1

Number of outcomes

3 × 4 = 12

2

Totals of 5 come from

(1, 4), (2, 3), (3, 2)

3

P(total is 5)

3/12 = ¼

4

Even totals: both odd or both even

(1,1), (1,3), (3,1), (3,3), (2,2), (2,4)

5

P(even)

6/12 = ½

ANSWER

P(total 5) = ¼ and P(even) = ½.

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EDEXCEL GCSE MATHS · FOUNDATION & HIGHER

Systematic Listing

Probability

Lesson 1 of 6

A system stops you missing outcomes or counting one twice.

Fix the first event

Write every second-event outcome beside it, then move to the next first-event outcome.

Use a table

Rows for one event, columns for the other.

Count

The total number of outcomes is the product of the number of outcomes of each event.

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Key Terms

Probability

Lesson 1 of 6

Outcome

One possible result of an event.

Event

One or more outcomes.

Equally likely

Each outcome has the same chance.

Sample space

The set of all possible outcomes.

Sample space diagram

A table or list showing every outcome.

Probability

A number from 0 to 1 showing how likely an event is.

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EDEXCEL GCSE MATHS · FOUNDATION & HIGHER

Probability

Lesson 1 of 6

YOUR TASK

Two-Dice Game

12 minutes

In a game, two dice are rolled and the smaller number is subtracted from the larger (or 0 if they are equal). Complete a table of differences. Which difference is most likely, and what is its probability?

1

Fill in a 6 by 6 table.

2

Count each difference.

3

Divide by 36.

WHAT A GOOD ANSWER SHOWS

The differences 0 to 5 occur 6, 10, 8, 6, 4 and 2 times out of 36. The most likely is 1, with probability 10/36 = 5/18.

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Can I...?

Probability

Lesson 1 of 6

Write a probability as a fraction.

List outcomes systematically.

Draw a sample space diagram.

Count successful outcomes.

Work out probabilities from a table.

Simplify probabilities.

Use 3 × 4 = 12 to count outcomes.

Avoid double-counting.

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EDEXCEL GCSE MATHS · FOUNDATION & HIGHER

Summary & Exam Focus

Probability = (successful outcomes)/(total outcomes).

Use a table or systematic list to find all outcomes.

The total number of outcomes for two events is the product.

Simplify your final fraction.

EXAM FOCUS

Two fair dice are rolled and their scores are added. Work out the probability that the total is 7. (3 marks)

Draw a 6 by 6 table. Count the 7s (there are 6) and divide by 36.

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EDEXCEL GCSE MATHS · FOUNDATION & HIGHER

Exam Practice: Combined Events

Probability

Lesson 1 of 6

Answer all questions. Show your working. · 25 minutes

Question 1 · 2 marks · Non-calculator

A fair coin is flipped and a fair dice is rolled. Write down the probability of getting a head and a 6.

Question 2 · 3 marks · Non-calculator

Two fair dice are rolled and the scores are added. Work out the probability that the total is 7.

Question 3 · 3 marks · Non-calculator

Spinner A has three equal sections numbered 1, 2 and 3. Spinner B has four equal sections numbered 1, 2, 3 and 4. Both spinners are spun…

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EDEXCEL GCSE MATHS · FOUNDATION & HIGHER

Exam Practice: Combined Events (cont.)

Probability

Lesson 1 of 6

Question 4 · 2 marks · Non-calculator

For the same two spinners, work out the probability that the total is even.

Question 5 · 3 marks · Non-calculator

Two fair dice are rolled and the scores are added. Work out the probability that the total is at least 10.

Question 6 · 2 marks · Non-calculator

Two fair dice are rolled. Work out the probability of getting a double (the same number on both dice).

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Question 1 · 2 marks · Non-calculator

Probability

Lesson 1 of 6

“

A fair coin is flipped and a fair dice is rolled. Write down the probability of getting a head and a 6.

HOW TO ANSWER IT

Command word: Non-calculator. Worth 2 marks, so plan before writing.

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EDEXCEL GCSE MATHS · FOUNDATION & HIGHER

Question 1 · mark scheme

Probability

Lesson 1 of 6

2 marks available. Award a mark for each point made.

12 outcomes

M1

1/12

A1

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Question 2 · 3 marks · Non-calculator

Probability

Lesson 1 of 6

“

Two fair dice are rolled and the scores are added. Work out the probability that the total is 7.

HOW TO ANSWER IT

Command word: Non-calculator. Worth 3 marks, so plan before writing.

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EDEXCEL GCSE MATHS · FOUNDATION & HIGHER

Question 2 · mark scheme

Probability

Lesson 1 of 6

3 marks available. Award a mark for each point made.

36 outcomes

M1

6 successful outcomes

M1

1/6

A1

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EDEXCEL GCSE MATHS · FOUNDATION & HIGHER

Question 3 · 3 marks · Non-calculator

Probability

Lesson 1 of 6

Spinner A has three equal sections numbered 1, 2 and 3. Spinner B has four equal sections numbered 1, 2, 3 and 4. Both spinners are spun and the two scores are added. Work out the probability that the total is 5. (3 marks)

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EDEXCEL GCSE MATHS · FOUNDATION & HIGHER

Question 3 · mark scheme

Probability

Lesson 1 of 6

3 marks available. Award a mark for each point made.

12 outcomes

M1

Three ways to make 5

M1

¼

A1

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EDEXCEL GCSE MATHS · FOUNDATION & HIGHER

Question 4 · 2 marks · Non-calculator

Probability

Lesson 1 of 6

“

For the same two spinners, work out the probability that the total is even.

HOW TO ANSWER IT

Command word: Non-calculator. Worth 2 marks, so plan before writing.

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EDEXCEL GCSE MATHS · FOUNDATION & HIGHER

Question 4 · mark scheme

Probability

Lesson 1 of 6

2 marks available. Award a mark for each point made.

6 successful outcomes

M1

½

A1

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EDEXCEL GCSE MATHS · FOUNDATION & HIGHER

Question 5 · 3 marks · Non-calculator

Probability

Lesson 1 of 6

“

Two fair dice are rolled and the scores are added. Work out the probability that the total is at least 10.

HOW TO ANSWER IT

Command word: Non-calculator. Worth 3 marks, so plan before writing.

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EDEXCEL GCSE MATHS · FOUNDATION & HIGHER

Question 5 · mark scheme

Probability

Lesson 1 of 6

3 marks available. Award a mark for each point made.

6 successful outcomes

M1

6/36

M1

1/6

A1

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EDEXCEL GCSE MATHS · FOUNDATION & HIGHER

Question 6 · 2 marks · Non-calculator

Probability

Lesson 1 of 6

“

Two fair dice are rolled. Work out the probability of getting a double (the same number on both dice).

HOW TO ANSWER IT

Command word: Non-calculator. Worth 2 marks, so plan before writing.

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EDEXCEL GCSE MATHS · FOUNDATION & HIGHER

Question 6 · mark scheme

Probability

Lesson 1 of 6

2 marks available. Award a mark for each point made.

6 doubles

M1

1/6

A1

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